Midpoint Rule Approximation with Six Rectangles

Math Problem Statement

Draw six rectangles with sample points as midpoints (M6). Then use the six rectangles to approximate the value of the integral from 0 to 12 of f(x) dx.

Solution

The problem is asking for an approximation of the integral $\int_0^{12} f(x) dx$ using the midpoint rule with six rectangles, denoted by $M_6$ .

Midpoint Rule:

The midpoint rule for approximating an integral $\int_a^b f(x) dx$ is given by: $M_n = \sum_{i=1}^{n} f(\text{midpoint}_i) \cdot \Delta x$ where:

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Math Problem Analysis

Mathematical Concepts

Integral Approximation
Midpoint Rule
Definite Integrals

Formulas

M_n = sum(f(midpoint_i) * Δx)
Δx = (b - a) / n

Theorems

Midpoint Rule

Suitable Grade Level

Grades 10-12

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